Loss Models: From Data to Decisions, 4th Edition

6.3 The negative binomial distribution

The negative binomial distribution has been used extensively as an alternative to the Poisson distribution. Like the Poisson distribution, it has positive probabilities on the nonnegative integers. Because it has two parameters, it has more flexibility in shape than the Poisson.

Definition 6.3 The probability function of the negative binomial distribution is given by

(6.2)

The binomial coefficient is to be evaluated as

While k must be an integer, x may be any real number. When x > k − 1, it can also be written as

which may be useful because ln Γ(x) is available in many spreadsheets, programming languages, and mathematics packages.

It is not difficult to show that the probability generating function for the negative binomial distribution is

From this it follows that the mean and variance of the negative binomial distribution are

Because β is positive, the variance of the negative binomial distribution exceeds the mean. This relationship is in contrast ...